Wafer shape is a geometric characteristic of a semiconductor wafer, which describes the position of the wafer's central plane surface in space. The bow, warp and other shape related parameters of semiconductor wafers must be within precise tolerances in order for wafers to be usable. The precision of a dimensional metrology (measurement) system must be tight enough to provide the required control over the quality of manufactured wafers.
The high accuracy metrology of test specimens, such as the topographic measurement of bow, warp, flatness, thickness etc. of such objects as semiconductor wafers, magnetic disks and the like, is impeded by the presence of noise in the output data. Depending on the inherent properties of the instrument and the environment, the data may have a noise content that displays larger peak to peak magnitude than the actual dimensions being measured. It is difficult to remove all sources of wafer vibration in a sensor based dimensional metrology system when the wafer moves between the sensors. The natural frequency of wafer vibration is of the order of tens to a few hundred Hertz, depending on wafer size and loading conditions, and the observed pattern of vibration has a spatial wavelength less than a few mm. If this noise is not removed, it directly affects the repeatability and reproducibility of the measurements of the system.
The measurements for wafer shape are typically taken at a plurality of points over the specimen surface. The positions of those points are not rigorously controlled between specimens. Therefore, the same data point may not be from the same exact location on each specimen tested by a particular metrology unit. This limits the usefulness of such noise elimination techniques as correlation analysis. Similarly the desire to process data for noise reduction from arbitrary shapes, particularly circular, reduces the attractiveness of high speed data systems such as Fast Fourier Transforms. Wafer shape is mostly a low spatial frequency characteristic. This makes it possible to remove vibration noise by using a low pass 2D spatial filter.
Convolution-based filters require a regular, evenly spaced data set that uses a priori information about the analytical continuation of the wafer shape beyond the wafer boundaries, e.g. the periodic behavior of the wafer shape. Because of this requirement for regular data and a priori information, conventional filters such as convolution techniques are not applicable for wafer shape vibration-noise removal. Fast Fourier transforms are an alternate high speed data processing method, but they are not well adapted to noise reduction processing from arbitrary non-rectilinear shapes, particularly circular shapes.
An analytical method for removing the noise content from metrology measurements of wafer specimens that accommodates the variability of data points is needed.